Measurable (α, β)-spaces with vanishing S-curvature

In this paper, we give a necessary and sufficient condition that an (alpha, beta)-space admits a measure mu with vanishing S-curvature everywhere. It is shown that the measure of such an (alpha, beta)-space must coincide with the Riemannian volume measure of alpha up to a constant multiplication. Moreover, the characterization of (alpha, beta)-spaces admitting a measure mu with weak isotropic S-curvature is given. (C) 2012 Elsevier B.V. All rights reserved.